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Suppose $G$ is a finitely generated group, $A$ is its finite set of generators. Lets denote the metric induced by the Cayley graph $Cay(G, A)$ on $G$ as $d$.

Suppose $\{X_i\}_{n = 0}^\infty$ is a sequence of i.i.d. random elements in $G$ and $\{Y_i\}_{n = 0}^\infty$ is another sequence of i.i.d. random elements in $G$ (the distributions of $X_0$ and $Y_0$ may still be different), that satisfy the following properties:

  1. $\forall n, m \in \mathbb{N}$ $X_n$ and $Y_m$ are mutually independant.

  2. $d(X_0, e)$ and $d(Y_0, e)$ both have finite second moments

Does it follow from that, that exists $r \in \mathbb{R}$ (the same for all $G$, $A$, $\{X_i\}_{n = 0}^\infty$ and $\{Y_i\}_{n = 0}^\infty$), such that $P(\inf_{g, h \in G}\overline{\lim_{n \to \infty}} \frac{d(\Pi_{j = 0}^nX_j, g)d(\Pi_{j = 0}^nY_j, h)}{n} \leq r) = 1$?

I managed to prove this fact only for finite groups (which is rather trivial):

$$\exists C \in \mathbb{N} \forall g, h \in G, d(g, h) < C$$ $$P(\inf_{g, h \in G}\overline{\lim_{n \to \infty}} \frac{d(X_n, g)d(Y_n, h)}{n} \leq \overline{\lim_{n \to \infty}} \frac{C^2}{n} \leq 0) = 1$$

This question was inspired by the theorem that says that random variables with finite second moment are uncorrelated.

First, I thought, that $r = 0$, but the counterexamples to that statement seem to be easily constructed for $G = \langle a \rangle_\infty$, $A = \{a\}$.

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  • $\begingroup$ What does i.i.d. stand for? $\endgroup$ – user1729 Apr 18 at 16:12
  • $\begingroup$ @user1729, "i.i.d" stands for "independent and identically distributed" $\endgroup$ – Yanior Weg Apr 18 at 16:14

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